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title: "A solid insulating sphere of radius \\(R\\) contains a non-uniform volume charge density \\(\\rho_1(r) = \\rho_0 \\left(\\dfrac{r}{R}\\right)\\), where \\(\\rho_0\\) is a positive constant and \\(r\\) is the radial distance from the center. Concentric with this sphere is a thick insulating spherical shell with inner radius \\(2R\\) and outer radius \\(3R\\). The thick shell contains a non-uniform volume charge density \\(\\rho_2(r) = -\\rho_0 \\left(\\dfrac{R}{r}\\right)\\). Which of the following expressions correctly gives the magnitude of the electric field \\(E(r)\\) in the region \\(2R < r < 3R\\) as a function of \\(r\\)?"
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url: "https://nerd-notes.com/ubq/118034/"
date_modified: "2026-08-04T08:03:16+00:00"
---

# A solid insulating sphere of radius \(R\) contains a non-uniform volume charge density \(\rho_1(r) = \rho_0 \left(\dfrac{r}{R}\right)\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the center. Concentric with this sphere is a thick insulating spherical shell with inner radius \(2R\) and outer radius \(3R\). The thick shell contains a non-uniform volume charge density \(\rho_2(r) = -\rho_0 \left(\dfrac{R}{r}\right)\). Which of the following expressions correctly gives the magnitude of the electric field \(E(r)\) in the region \(2R < r < 3R\) as a function of \(r\)?

A solid insulating sphere of radius \(R\) contains a non-uniform volume charge density \(\rho_1(r) = \rho_0 \left(\dfrac{r}{R}\right)\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the center. Concentric with this sphere is a thick insulating spherical shell with inner radius \(2R\) and outer radius \(3R\). The thick shell contains a non-uniform volume charge density \(\rho_2(r) = -\rho_0 \left(\dfrac{R}{r}\right)\). Which of the following expressions correctly gives the magnitude of the electric field \(E(r)\) in the region \(2R < r < 3R\) as a function of \(r\)?

![Cross-sectional view of concentric spherical regions centered at the origin. At the center is a solid shaded circle of radius R labeled with charge density \rho_1(r) = \rho_0(r/R). Surrounding it is a region of empty space extending from radius R to 2R. Next is a thick spherical shell extending from inner radius 2R to outer radius 3R, filled with hatch lines and labeled with charge density \rho_2(r) = -\rho_0(R/r). A dashed circular Gaussian surface of radius r sits in the region between 2R and 3R. A radial arrow from the origin to the dashed circle is labeled r. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830595-dH5L3S.jpg)

- **A.** \(\dfrac{\rho_0 R}{4\varepsilon_0} \left( \dfrac{8R^2}{r^2} - 2 \right)\)
- **B.** \(\dfrac{\rho_0 R}{4\varepsilon_0} \left( \dfrac{28R^2}{3r^2} - 2 \right)\)
- **C.** \(\dfrac{\rho_0 R}{4\varepsilon_0} \left( \dfrac{17R^2}{r^2} - 4 \right)\)
- **D.** \(\dfrac{\rho_0 R}{4\varepsilon_0} \left( \dfrac{9R^2}{r^2} - 2 \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118034/*